Engineered quantum mechanics with nano-electronics Robert Johansson - - PowerPoint PPT Presentation

Engineered quantum mechanics with nano-electronics

Collaborators:

• F. Nori (RIKEN), G. Johansson & P. Delsing (Chalmers),
• C. Wilson (U. Waterloo), J. Stehlik & J. Petta (Princeton)
• P. Nation (Korea U.), N. Lambert (RIKEN)

Robert Johansson (JSPS/RIKEN)

Content

• Brief review to quantum electronics
• Overview of research
• Summary of selected research projects
• Dynamical Casimir effect

– QFT/Quantum optics with superconducting circuits

• Landau-Zener-Stuckelberg interferometry

– Atomic physics with semiconducting quantum dots

• QuTiP: Framework for computational quantum dynamics

– Computational physics

• Conclusions

Review of superconducting circuits

from qubits to on-chip quantum optics

2000 2005 2010

NEC 1999

qubits qubit-qubit

qubit-resonator

resonator as coupling bus

high level of control

• f resonators

Delft 2003 NIST 2007 NEC 2007 NIST 2002 Saclay 2002 Saclay 1998 Yale 2008 Yale 2011 UCSB 2012 UCSB 2006 NEC 2003 Yale 2004 UCSB 2009 UCSB 2009 ETH 2008 ETH 2010 Chalmers 2008

Review of quantum electronics

current status of the field

Implementable using superconducting and semiconducting nanoelectrical devices:

• Few-level quantum systems (qubits, artificial atoms)
• Controllable by design and/or in-situ tunable
• Resonators (cavities)
• Linear and nonlinear
• Frequency tunable
• Flexible coupling (electric, magnetic, position)
• Strong and ultrastrong coupling regimes demonstrated
• Hybrid devices: quantum systems of different nature can be coupled
• Various readout schemes available

With these building blocks, a wide range of quantum phenomena can be realized and simulated in engineered nanoelectronical devices.

Overview of research

Dynamical Casimir effect

In collaboration with F. Nori (RIKEN), G. Johansson, P. Delsing (Chalmers),

• C. Wilson (Waterloo)

PRL 2009 PRA 2010 Nature 2011 PRA 2013

Quantum field theory: vacuum effects

Casimir force (1948)

Experiment: Lamoreaux (1997)

Hawking Radiation Dynamical Casimir effect Lamb shift

(Lamb & Retherford 1947)

Unruh effect A review of quantum vacuum effects: Nation, Johansson, Blencowe and Nori, RMP (2012). Examples of physical phenomena due to quantum vacuum fluctuations (with no classical counterparts).

A mirror undergoing nonuniform relativistic motion in vacuum emits radiation In general: Rapidly changing boundary conditions of a quantum field can modify the mode structure

• f quantum field nonadiabatically, resulting in

amplification of virtual photons to real detectable photons (radiation). Examples of possible realizations:

• Moving mirror in vacuum (mentioned above)
• Medium with time-dependent index of refraction

(Yablanovitch PRL 1989, Segev PLA 2007)

• Semiconducting switchable mirror by laser irradiation

(Braggio EPL 2005)

• Our proposal:

Superconducting waveguide terminated by a SQUID

(PRL 2009, PRA 2010, experiment Wilson Nature 2011, review Nation RMP 2012, PRA 2013)

Dynamical Casimir effect cartoon Moore (1970), Fulling (1976) Reviews: Dodonov (2001, 2009), Dalvit et al. (2010)

The dynamical Casimir effect

Case Frequency (Hz) Amplitude (m) Maximum velocity (m/s) Photon production rate (#photons/s)

Moving a mirror by hand “handwaving” 1 1 1 ~ 1e-18 Mirror on a nano-mechanical

• scillator

1e+9 1e-9 1 ~1e-9

The problem with massive mirrors

Examples of DCE photon production rates for some naive single mirror systems

Lambrecht PRL 1996.

Photon production rate: The very low photon-production rate makes the DCE difficult to detect experimentally in systems with mechanical modulation of the boundary condition (BC). → need a system which does not require moving massive objects to the change BC.

Frequency tunable resonators

SQUID-terminated transmission line:

See also: Yamamoto et al., APL 2008 Kubo et al., PRL 105 140502 (2010) Wilson et al., PRL 105 233907 (2010)

Sandberg et al., APL 2008 Palacios-Laloy et al., JLTP 2008 Castellanos-Beltran et al., APL 2007

Wallquest et al. PRB 74 224506 (2006)

Superconducting circuit for DCE

The boundary condition (BC) of the coplanar waveguide (at x=0):

• is determined by the SQUID
• can be tuned by the applied magnetic flux though the SQUID
• is effectively equivalent to a “mirror” with tunable position (1-to-1 mapping of BC)

No motion of massive objects is involved in this method of changing the boundary condition. ... ... ...

PRL 2009

Superconducting circuit for DCE

... ... ... The “position of the effective mirror” is a function of the applied magnetic flux: Coplanar waveguide: Equivalent system:

Superconducting circuit for DCE

Harmonic modulation of the applied magnetic flux:

• BC identical to that of an oscillating mirror
• produces photons in the coplanar waveguide (dynamical Casimir effect)

... ... ... Coplanar waveguide: Equivalent system: A perfectly reflective mirror at distance

Circuit model → Boundary condition

Circuit model:

• Hamiltonian:
• We assume that the SQUID is only weakly excited (large plasma frequency)
• The equation of motion for gives the boundary condition for the

transmission line:

Perturbation solution for sinusoidal modulation: Now, any expectation values and correlation functions for the output field can be calculated: For example, the photon flux in the output field for a thermal input field:

Input-output result for oscillating BC

Reflected thermal photons Dynamical Casimir effect ! Reflected thermal photons

Predicted output photon-flux density vs. mode frequency: → broadband photon production below the driving frequency thermal Radiation due to the dynamical Casimir effect

Red: thermal photons Blue: analytical results Green: numerical results T emperature:

• Solid: T = 50 mK
• Dashed: T = 0 K

Example of photon-flux density spectrum

(plasma frequency) (driving frequency)

Photon production rates

Lambrecht et al., PRL 1996.

Photon production rate: Case Frequency (Hz) Amplitude (m) Maximum velocity (m/s) Photon production rate (# photons / s) moving a mirror by hand 1 1 1 ~1e-18 nano-mechani cal oscillator 1e+9 1e-9 1 ~1e-9 SQUID in coplanar waveguide 18e+9 ~1e-4 ~2e6 ~1e5

DCE in a cavity/resonator setup

PRA 2010

Symmetric double-peak structure when ω1 + ω2 = ωd Photons in the ω1 and ω2 modes are correlated.

Open waveguide case: single broad peak

ω1 ω2

Example of two-mode squeezing spectrum

• DCE generates two-mode squeezed states (correlated photon pairs)
• Broadband quadrature squeezing

Advantages:

• Can be measured with

standard homodyne detection.

• Photon correlations at different

frequencies is a signature of quantum generation process. Solid lines: Resonator setup Dashed lines: Open waveguide

The experiment

Schematic Experiment

SQUID Wilson (Nature 2011) PRL 2009, PRA 2010

See also: Lähteenmäki et al., PNAS (2012)

The experiment

Schematic Experiment

SQUID

See also: Lähteenmäki et al., PNAS (2012)

Wilson (Nature 2011) PRL 2009, PRA 2010

Measured reflected phase

Testing the tunability of the effective length: Measurement of the phase acquired by an incoming signal that reflect off the SQUID as a function of the externally applied static magnetic field. The reflected phase is directly related to the effective “electrical length” of the SQUID.

(Nature 2011)

Measured photon-flux density

Fix the pump frequency and vary the analysis frequency: We expect to see a symmetric spectrum around zero detuning from half the pump frequency.

Fix this parameter Measure in this range of frequencies

Measured photon-flux density

Broadband photon production is observed, and the measured spectrum is clearly symmetric around the half the pump frequency (zero digitizer detuning in figure below).

Measured photon-flux density

Broadband photon production is observed, and the measured spectrum is clearly symmetric around the half the pump frequency (zero digitizer detuning in figure below).

Averaged photon flux in the ranges indicated above

Measured photon-flux density

Broadband photon production is observed, and the measured spectrum is clearly symmetric around the half the pump frequency (zero digitizer detuning in figure below).

Averaged photon flux in the ranges indicated above

Measured photon-flux density

Broadband photon production is observed, and the measured spectrum is clearly symmetric around the half the pump frequency (zero digitizer detuning in figure below).

Photon flux vs pump power for the cut indicated above

Measured two-mode correlations and squeezing

Voltage quadratures: Symmetrically around half the driving frequency: Strong two mode squeezing is observed (only) if → strong indicator for photon-pair production. Also, single-mode squeezing is not observed, as expected from the dynamical Casimir effect theory (where only two-photon correlations are created).

No correlations without pump signal

The correlations vanish when:

• the pump is turned off
• the two analysis frequencies does not sum up to the pump frequency:

The parasitic cross-correlations intrinsic to the amplifier are very small. Compare to ~25% → squeezing in the figure on the Previous page.

Theory: Quantum-classical indicators

• Two-photon correlations and two-mode

squeezing are nonclassical, but what about the entire field state including of thermal noise?

• Use a nonclassicality test based on the

Glauber-Sudarshan P-function:

• For DCE in our circuit:

(See e.g. Miranowicz PRA 2010) (good for cross-quadrature squeezing)

PRA 2013

Overview of research

Landau-Zener-Stuckelberg interferometry

in a double-quantum dot device

In collaboration F. Nori (RIKEN) and J. Stehlik and J. Petta (Princeton)

Ashhab, Johansson, Zagoskin, Nori PRA 2007 Stehlik et al. PRB 2012

The Landau-Zener problem

A two-level system is swept through an avoided-level crossing.

• Fast passage

→ nonadiabatic transition from ground state to excited state

• Slow passage → adiabatically following the instantaneous ground state

Landau-Zener transition probability:

The Landau-Zener-Stuckelberg problem

Repeated, cyclic Landau-Zener transitions.

• Transitions are only possible near the avoided-level crossing
• Away from the crossing the wavefunction accumulates a phase: Stuckelberg phase
• Constructive or destructive interference can occur at successive avoided-level crossings
• The interference condition depends on:
• The late-time occupation probabilities are dominated by interference dynamics.

The Landau-Zener-Stuckelberg problem

Repeated, cyclic Landau-Zener transitions.

• Transitions are only possible near the avoided-level crossing
• Away from the crossing the wavefunction accumulates a phase: Stuckelberg phase
• Constructive or destructive interference can occur at successive avoided-level crossings
• The interference condition depends on:
• The late-time occupation probabilities are dominated by interference dynamics.

Steady state occupation probability

The Landau-Zener-Stuckelberg problem

Repeated, cyclic Landau-Zener transitions.

• Transitions are only possible near the avoided-level crossing
• Away from the crossing the wavefunction accumulates a phase: Stuckelberg phase
• Constructive or destructive interference can occur at successive avoided-level crossings
• The interference condition depends on:
• The late-time occupation probabilities are dominated by interference dynamics.

destructive constructive Steady state occupation probability

The single-electron charge qubit

Stehlik et al., Phys. Rev. B 86, 121303(R) 2012 A semiconducting double quantum dot device:

Quantum states:

Transport measurement

Driving power

• With an applied driving field, transport current is observed for biases which
• therwise would be in the blockade regime.
• A characteristic Landau-Zener-Stuckelberg interference pattern is observed.

Transport current

Model for the transport current

We model the transport measurements and the current through the device using:

• Three quantum states: occupied left dot (1,0), right dot (0,1), and both dots empty (0,0)
• A time-dependent Lindblad master equation with
• incoherent tunneling from leads to dots:
• incoherent thermal relaxation/excitation between the dots:
• inter-dot dephasing:
• coherent tunneling between the dots and energy bias:

Detuning and driving-power dependent

Model results for the transport current

• Good agreement between experimental transport measurements and the predictions of the

theoretical model.

• Regions where the current is suppressed for large driving power correspond to destructive LZS

interference, or equivalently coherent destruction of tunneling (CDT).

Measurements Numerical model Floquet quasienergies

Charge sensing measurement at 15 GHz

• Charge sensing is an alternative to transport measurement.
• It does not disturb the coherent dynamics of the two qubit states as much and is

therefore closer to the original LZS problem, allowing higher-order photon processes to be observed.

Charge sensing measurement at 9 GHz

With charge-sensing measurements, up to 17-photon resonances were observed.

Overview of research

QuTiP: Quantum toolbox in Python

Framework for computational quantum dynamics In collaboration with P.D. Nation (Korea Univ.) and F. Nori (RIKEN)

http://qutip.org

Why QuTiP?

The rapidly increasing complexity of quantum electronics requires new improved analysis tools

2000 2005 2010

NEC 1999

qubits qubit-qubit

qubit-resonator

resonator as coupling bus

high level of control

• f resonators

Delft 2003 NIST 2007 NEC 2007 NIST 2002 Saclay 2002 Saclay 1998 Yale 2008 Yale 2011 UCSB 2012 UCSB 2006 NEC 2003 Yale 2004 UCSB 2009 UCSB 2009 ETH 2008 ETH 2010 Chalmers 2008

About QuTiP

Publications:

• Comp. Phys. Comm. 183, 1760 (2012)
• Comp. Phys. Comm. 184, 1234 (2013)

Users:

• Thousands of downloads since latest release

in March 2013

• Used at hundreds of research institutes and

universities around the world

• > 1000 unique visitors every month to the

QuTiP project web page http://qutip.org

Organization of the QuTiP software

~30000 lines of Python code, ~50 modules, ~20 quantum dynamics solvers

Time evolution Visualization Gates Operators States Quantum objects Entropy and entanglement

What is QuTiP used for?

Used to solve a wide range of problems in different fields

Quantum dots Superconducting circuits Quantum optics Lasing Cavity/Circuit QED Quantum states of light Phase transitions Quantum information and Communication Decoherence

Conclusions

• Quantum electronics: using superconducting and semiconducting nanoelectronics
• Powerful platform for engineered quantum mechanics, including simulations and

implementations of quantum phenomena from quantum optics, atomic and molecular physics, quantum field theory, chemistry, and perhaps even biology.

• Examples of interdisciplinary quantum project:
• Dynamical Casimir effect in superconducting circuits

–

First experimental demonstration of the effect possible due to a quantum electronics realization

• Landau-Zener-Stuckelberg interferometry with a semiconducting quantum dot

–

“Amplitude spectroscopy” method for characterizing quantum dots inspired by atomic physics

• QuTiP: versatile framework for quantum dynamics simulation

–

Potentially applicable in many fields of quantum physics

• Current and future work
• Circuit QED with graphene quantum dots
• Producing nonclassical states in multimode nanomechanical resonators
• Simulating quantum phenomena in superconducting/semiconducting electrical circuits